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<b>let </b><font color="Maroon" title="c1">E</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="rewrite2.html#NM1" title="REWRITE2:NM.1">semi-Thue-system</a> of <font color="Maroon" title="c1">E</font>;<br/><b>let </b><font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">t</font> be    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> ;<br/>







<b>assume </b><a NAME="E1:61"/><i><font color="Green" title="E38">A1</font></i>: 
( <font color="Maroon" title="c2">S</font> is   <a href="rewrite2.html#NM2" title="REWRITE2:NM.2">Thue-system</a> of <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon" title="c3">s</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c2">S</font> )
 ;<br/>

<a NAME="E2:61"/><b>then </b>
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font>
 
<b>by </b><i><a class="ref" href="rewrite2.html#D7" target="_self" title="REWRITE2:def.7">Def7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">p</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E4:61"/><i><font color="Green" title="E39">A2</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D3" title="REWRITE1:def.3">REWRITE1:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">q</font> =  <a href="finseq_5.html#K3" title="FINSEQ_5:func.3">Rev</a> <font color="Maroon" title="c5">p</font>;<br/>
<a NAME="E5:61"/>
 <a href="finseq_5.html#K3" title="FINSEQ_5:func.3">Rev</a> <font color="Maroon" title="c5">p</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of <span class="p1">(<span class="default"><a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> 
 
<b>by </b><i><a class="ref" href="rewrite1.html#T10" title="REWRITE1:th.10">REWRITE1:10</a></i>;<br/>
<a NAME="E6:61"/><b>then </b><i><font color="Green" title="E40">A3</font></i>: 
 <a href="finseq_5.html#K3" title="FINSEQ_5:func.3">Rev</a> <font color="Maroon" title="c5">p</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="rewrite2.html#E1:61"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="rewrite2.html#T25" target="_self" title="REWRITE2:th.25">Th25</a></i>;<br/>
<a NAME="E7:61"/><i><font color="Green" title="E41">A4</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_5.html#K3" title="FINSEQ_5:func.3">Rev</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font>
 
<b>by </b><i><a class="txt" href="rewrite2.html#E4:61"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a></i>;<br/>
<a NAME="E8:61"/>
<span class="p1">(<span class="default"><a href="finseq_5.html#K3" title="FINSEQ_5:func.3">Rev</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font>
 
<b>by </b><i><a class="txt" href="rewrite2.html#E4:61"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a></i>;<br/>
<a NAME="E9:61"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_5.html#K3" title="FINSEQ_5:func.3">Rev</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K3" title="FINSEQ_5:func.3">Rev</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/>
<a NAME="E10:61"/><b>then </b>
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c3">s</font>
 
<b>by </b><i><a class="txt" href="rewrite2.html#E6:61"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="rewrite2.html#E7:61"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="rewrite1.html#D3" title="REWRITE1:def.3">REWRITE1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E11:61"/>
<font color="Maroon" title="c4">t</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="rewrite2.html#D7" target="_self" title="REWRITE2:def.7">Def7</a></i>;<br/>


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